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Generalized (PHP) pigeon hole principle
If you have N objects in K boxes then there is some box with r = [N/K] objects
Note: "x" is the ceiling function also called the least integer function.
Example 325 students in MAT 226, 5 possible gr
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6.2 Pigeon-hole principle (PHP)
- h boxes
- k+1 objects
Some boxes must have at least two objects.
Example
6 objects 5 boxes
See picture
Example Birthdays
12 months 13 people
Example
How many people are required to guarantee a day of the y
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b) How many socks must be selected to guarantee 4 of the same color
2(4-1) + 1 = 7
c) How many socks must be drawn to guarantee 4 black socks
Not PHP 2(4-1) + 1 = 16?
6.3 Permutations and Combinations
Example
Let S = cfw_Jordan, Ann, Adam
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Definition: A one-to-one function has for each element in the range only one
mapping from the domain.
Example
See picture
Example
How many functions f: A -> B are possible?
n * n n = nm
Example
How many one to one functions are possible if
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Section 6.1 Counting
Sum Rule: if task 1 can be performed in 'n1' ways and task 2 in 'n2' ways, then
there are 'n1 + n2' ways for the procedure.
Example
4 store brands
2 name brands
How many ways are there to choose? Well 4 + 2 = 6 so A: 6
Part 1
Part 2
Part 3
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Part 1
Recurrence Relations
Conditional Probability
Generalized Inclusion/ Exclusion
P(knowing 'union' problem)
P(knowing)
an+1 = 2an
rn+1 = 2rn
rn = r n
r=2
Part 2
Example
A = cfw_1, 2, 3
B = cfw_3, 4, 5
5 'line-Epsil
Example NAU phones start with 523 or 522 and not 911
10,000,000 - 5
9,999,995
|P(S)| = 2|S|
Proof:
Let S = cfw_S1, S2, , Sn
|S| = n
Choices for subsets are
2 * 2 * * 2 = 2n = 2|S|
Example
1.
How many possible 4 letter strings are there?
26^4
How many with
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Definition: The universal set U is the set of all elements under consideration.
Example U = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9, 10
A = cfw_1, 2, 3
B = cfw_ 3, 4, 5
Definition: The complement of the set A, denoted A or AC, is the set of all ele
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Exactly one 'C'
1 25 25 25
25 1 25 25
25 25 1 25
25 25 25 1
= 4 * 25
Inclusion
Exclusion: |A 'union' B| = |A| + |B| - |A 'intersection' B|
Functions
Definition: A function is a mapping of elements in set A to set B. Each element in
A is as
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Mat 226
First Time Discrete Math
Sec 2.1/ 2.2 sets
Definition: A set is a collection of objects called elements
Notations/ definitions
Use capital letters for sets
'Epsilon' is an element of
'line-through-Epsilon' is not an element of